Assessment of Loads and Structural Integrity of Ships ...

30th Symposium on Naval Hydrodynamics Hobart, Tasmania, Australia, 2-7 November 2014

Assessment of Loads and Structural Integrity of Ships in Extreme Seas Jens Ley, Ould el Moctar (University of Duisburg-Essen), Jan Oberhagemann, Thomas E. Schellin (DNV GL) ABSTRACT For ships advancing with forward speed in extreme seas, numerical methods are proposed that couple solvers of the Reynolds-Averaged Navier-Stokes (RANS) equations with solvers of rigid and flexible body displacements. The methods simulate nonlinear wave hydrodynamics and predict the corresponding nonlinear wave-induced ship motions and loads. Two containerships are numerically investigated in random irregular long crested waves, and deterministic wave sequences. Comparisons with experimental measurements show good agreements. Different wave models are presented, including second order Stokes waves and nonlinear wave fields obtained from the solution of nonlinear Schrödinger equations (NLS). Simulations in random irregular waves provide short-term ship response probability distributions in sea state conditions relevant for design loads. INTRODUCTION In recent years, several accidents at sea were related to exceptionally high and steep waves, so-called rogue waves. These accidents as well as in-situ wave measurements attracted public attention. Terms such as freak waves became common parlance, suggesting a mythological and scary nature of exceptional waves. Contrary to earlier assumptions, the risk for ships to encounter extreme waves is not negligible, and a ship’s structural integrity and stability may be endangered when the master cannot avoid these extreme seas. These findings call for research in order to understand the nature of extraordinary large waves and their impact on ship structures. Although available marine accident statistics do not indicate extreme wave events to be a major risk to life and property at sea, a sound understanding of the related physics is required to draw profound conclusions. Accordingly, several research projects were dedicated to this topic, such as the recently terminated EUfunded project EXTREME SEAS (2013). Numerous numerical approaches exist to assess wave-ship interaction, ranging from strip theory methods to boundary element methods (BEMs) to field methods based on the solution of

Reynolds-Averaged Navier-Stokes (RANS) equations. The latter are best suited when nonlinearities significantly affect the wave-induced ship responses. Furthermore, they are becoming more accessible to the engineering community to assess wave-structure interaction. Indeed, RANS methods are the preferred choice to model particularly complicated situations (ISSC, 2012). We present approaches that couple field methods with solvers of ship motions and deformations to simulate nonlinear seaways and to predict corresponding nonlinear wave-induced ship motions and loads. Ship dependent relevant scenarios were identified and analyzed using different wave models, e.g., second order Stokes waves and nonlinear wave fields obtained from the solution of nonlinear Schrödinger equations (NLS) derived by Zhakarov (1968). The corresponding code was provided by the University of Torino. We also look at the associated wave propagation in these wave fields (Ley et al., 2013). The nonlinear Schrödinger equation has a number of exact analytical solutions, known as breather solutions, which are prototypes of rogue waves (Osborne et al., 2000). Such waves are relevant because they may emerge spontaneously from a random sea state provided the spectrum is sufficiently narrow and waves, on average, are sufficiently steep (Onorato et al., 2001). The first task consisted of incorporating hydroelastic effects in the numerical simulation of extreme ship responses. Basic hull girder vibrations and local structural deformations had to be accounted for together with wave-induced ship loads occurring in caused wave groups. For this purpose, a numerical method was developed, suitable for extreme wave generation and for coupling of a free surface flow solver with a finite element based structural solver. Regarding simulation of ship responses, three aspects were addressed. First, a Timoshenko beam model of the ship structure was incorporated into a RANSE/Euler solver through user-defined routines that accounted for two-way coupling between hydrodynamic forces and six-degrees-offreedom rigid body motions and elastic hull deformations (Brunswig and el Moctar, 2004; Oberhagemann and el Moctar, 2007). The code was

systematically validated against representative model test measurements (Oberhagemann and el Moctar, 2011). Second, a one-way coupling method, including a three-dimensional model of the ship structure for local structural analyses, was introduced. Regarding simulations of wave-induced loads, relevant sea conditions were identified leading to extreme ship responses. Comparisons between model test measurements and numerical simulations validated the newly developed tools. Not only regular waves, but also irregular long crested seaways and deterministic wave trains that incorporated rogue waves were numerically investigated and compared with experimental measurements. One of the objectives was to obtain statistical information of the probability of occurrence of extreme waves and the corresponding ship responses. Here, this was done for two sample ships. Finally, we determined conditioned wave sequences based on the so-called Most Likely Response Wave (MLRW) concept (Dietz et al., 2004). These induced a linear vertical bending moment response at target time t0, corresponding to the long-term expected maximum according to spectral moment statistics. Subsequent RANS simulations in the wave sequences provided nonlinear corrections to the linear responses. RANS FIELD METHODS The RANS equations solvers COMET and interDyMFoam, of the CFD toolbox OpenFOAM, solve the conservation equations for mass and momentum in their integral form. The fluid is assumed to be viscous. The solution domain is subdivided into a finite number of control volumes that may be of arbitrary shape. The integrals are numerically approximated using the midpoint rule. Mass fluxes through the cell faces are taken from the previous iteration, following a simple Picard iteration approach, (Ferziger & Peric, 2008). To obtain a dedicated equation for pressure, the mass conservation equation is converted into a pressure correction equation. Code COMET achieves an implicit coupling between pressure and velocity using the Semi-Implicit Pressure Linked Equations (SIMPLE) algorithm, while code interDyMFoam (2014) implements a hybrid PIMPLE approach that combines SIMPLE with the PISO (Pressure Implicit with Splitting of Operators) algorithm. For simulation of the free-surface flows around floating bodies, both RANS methods implement a Eulerian multiphase formulation. An additional transport equation solves for a scalar flow quantity, α, which represents the volume fraction of

the fluids involved, in this case water and air. Scalar α determines the intensive fluid parameters, so the conservation equations are solved for an effective fluid. E.g., the effective fluid density ρeff reads

eff    water  (1   )  air

(1)

Special discretization schemes are required to reduce numerical smearing and to retain a sharp interface between water and air. COMET provides the HRIC scheme (High Resolution Interface Capturing), whereas OpenFOAM uses the MULES algorithm (Multidimensional Universal Limiter with Explicit Solution) and an additional compressive convective term. Both interface capturing methods implicitly account for nonlinear free surface effects such as overturning and breaking waves, buoyancy effects of trapped air, and wave-wave-interaction (el Moctar et al., 2011). Such free surface features should be considered to predict slamming pressures accurately. TIMOSHENKO BEAM APPROACH The numerical method to analyze flexible hull girder responses relied on the Timoshenko beam approach, combined with RANS codes Comet and interDyMFoam. These solvers were coupled in an iterative two-way coupling scheme with the fluid solvers. The RANS solvers computed pressure forces and friction induced shear forces acting on the ship hull, which were input to the six-degree-of-freedom ship motion solver and the structural solver. Finite element nodal displacements were solved with the structural solver and transferred back to the ship motion solver and mapped onto the CFD mesh. A grid morphing method obtained grid deformations. This scheme was iterated in each time step until convergence, with under-relaxation applied to all solvers. Hence, at the end of a time step, a converged and consistent solution for the flow, the ship motions, and the elastic deformations was achieved. The ship hull girder was modeled by a finite element Timoshenko beam. The beam elements accounted for planar bending and shear deformation with third order shape functions and two degrees of freedom per node, i.e. the vertical deformation, u, and the rotation about the transverse axis of the ship, ψ. Time integration used the second order implicit Newmark approach (Oberhagemann & el. Moctar, 2012). 3D FINITE ELEMENT MODEL APPROACH Combining the commercial RANS solver StarCCM+ and the structural solver Ansys (2009), a

second numerical method was developed to account for hull girder flexibility. This investigation was based on a 3D Finite Element (FE) structural model of one investigated ship. 3D FE models are required for the assessment of local structural responses and for complex global loads, e. g. torsion with constrained warping. The Mesh-based parallel Code Coupling Interface (MpCCI) established communication between the solvers, allowing exchanging and mapping of selected quantities between two different meshes and solvers. This framework seemed suitable to account for global and local hull girder flexibility. However, further enhancements were required to balance residual forces resulting for example from rounding errors. Based on the Ansys Parametric Design Language (APDL), user defined routines were developed and incorporated in the run loops of coupled simulations to balance residual forces and moments and to counteract an additional acceleration field of the rigid ship. These enhancements were applied for three different coupling approaches: 1. 2.

3.

One-way coupling, where only pressure forces were transmitted Enhanced one-way coupling, where pressure forces were transmitted and the structure interacted with fluid elements Weak two-way coupling, where pressure forces and nodal deformations were exchanged

While the first approach was applied for the assessment of local green water loads and corresponding structural responses (Ley et al. 2013), the second was used to analyze global hull girder loads. Here, acoustic fluid elements attached to the FE structural model compensated the disregard of hydrodynamic forces in reaction to structural accelerations, i.e. the fluid elements served to account for added mass effects. The third approach exchanged loads and deformations once per time step (weak coupling). Additional stabilizing mechanisms had to be implemented to improve the convergence of RANS simulations, comprising subcycling, ramping of pressures to reduce the initial excitation of the FE model, ramping of structural damping, under relaxation of nodal deformations and, as water was assumed compressible, reduction of the speed of sound and the associated pressure disturbances. ADDED MASS MODELING The long-term goal of load assessment of a flexible ship hull operating in waves is to establish a two-way implicit coupling method using a three-dimensional

FE model and performing transient analyses for the computational fluid dynamics simulations and the computational structural dynamics simulations. Nonlinear hydrodynamic effects were to be captured by the RANS solver and nonlinear structural behavior (plate buckling, structural failure, etc.) by the structural solver. However, this turned out to be too formidable a task and, therefore, simplified approaches were developed, including explicit coupling techniques to reduce simulation time. Within the scope of this work two different methods accounting for global hull girder vibrations were used, namely, a field method for fluid dynamics and a three-dimensional FE model for structural dynamics. While the one-way coupling method neglected the effect of added masses for elastic deformations, hydrodynamic restoring as well as hydrodynamic damping forces and moments, the weak two-way coupling technique included these effects in principle. However, stability and convergence problems in the fluid dynamic solution caused by this coupling technique made it difficult to find a generic solution. To overcome these shortcomings, the enhanced one-way coupling procedure was introduced (Ley and el Moctar, 2014). This method, based on the theory of acoustic waves, treats the generation, propagation, absorption, and reflection of sound pressure waves in the fluid medium. The acoustic elements compensate the disregard of hydrodynamic forces in reaction to structural acceleration (added mass effects). These forces cause a significant decrease of natural frequencies in wetted condition, compared to those in air. In addition, hydrodynamic damping may be determined. Added mass coefficients depend on hull geometry, frequency of vibration, water depth, and forward speed of the ship. Instead of using constant added masses valid for only one frequency, the use of acoustic elements is a more generic and consistent approach. Reaction forces are modeled correctly, irrespective of the excitation frequency. The often used method to describe the flow is based on the velocity potential satisfying the Laplace equation and additional boundary conditions. The acoustic wave equation 1 2 p  2 p  0 c 2 t 2

(2)

governs, with c = (k/ρ0)1/2 denoting the speed of sound, k the bulk modulus of fluid, ρ0 the mean fluid density, p(x,y,z,t) the acoustic pressure, and t the time. This equation results from the fluid momentum

and continuity equations (1) and (2), assuming the flow is incompressible, inviscid, initially at rest, and the fluid is of uniform density. Multiplying the wave equation by a virtual change in pressure, δp, and integrating over the volume of the domain yields:  1 2 p 2 V c 2 t 2  p dV  V  p  p dV  S np  p dS

(3)

eigensolver for asymmetric matrices, via a full harmonic response analysis, and via a full transient structural analysis. Here, modal analyses were performed to determine the natural frequencies of wetted hull surfaces, while full transient analyses were used during the coupled simulations. WAVE MODELING

To solve the unknown parameters p and u , shape functions from the finite element formulation are applied:

The wave energy of irregular wave processes is described with the spectral energy density distribution, S, as function of wave frequency, ω. Theoretical models provide semi-empirical correlations S(ω). The most common are the PiersonMoskowitz spectrum which only depends on wind speed, the modified Pierson-Moskowitz spectrum which depends on significant wave height and zero up-crossing period, and the JONSWAP spectrum for limited fetch and wind duration. The International Association of Classification Societies (IACS) recommends the modified Pierson-Moskowitz spectrum for wave load predictions of ships. It corresponds to a JONSWAP spectrum with a peak enhancement factor of γ = 1.0. Wave elevation, wave velocity field, and pressure field are imposed at the fluid domain boundaries. In case of irregular waves, the wave process is discretized by superposition of n linear harmonic component waves according to Airy theory. The surface elevation of unidirectional waves reads

      p  N  pe , u  N ' ue ,  p  N   pe

 ( x, t )   Ai cos(ki x  i t )

with V denoting the volume of the domain, S the surface where the derivative of the surface is applied,  and n the unit normal to the interface S. The normal pressure gradient of the fluid and the normal acceleration of the structure at the interface are linked:    2u n p    2 t

(4)



Here, u is the nodal displacement Substituting (4) into (3) yields

vector.

 1 2 p  2u 2  p dV   p  p dV  2 2 2  c t  S t  p dS  0 (5) V V 

n

(6)

(9)

i

For δp being non-zero and moving terms which do not vary over the element in front of the integral sign yields 1 c2

2

N V

     dV pe    2 N dV pe   0  nNN ' dS u  0 V

S

(7) This equation can be expressed in matrix form in notation similar to the motion equation (Ley & Schellin, 2013). To account for energy loss at the absorbing boundary surface, a fluid damping term can be added:



2

N c 

 dS  p e

(8)

S

with ß = ra/ρ0c, where ra is the absorption rate at the boundary. The total mass and stiffness matrices are asymmetric and can be solved via a modal analysis using the

with surface elevation ζ(x,t), component wave amplitudes Ai, wave frequencies ωi,, and corresponding wave numbers ki = ωi²/g. Velocity and pressure field are composed accordingly from component waves. This kind of component wave superposition neglects wave-wave interaction and, thus, introduces a certain error at the boundaries. Inside the fluid domain, waves propagate according to the discretized Navier-Stokes equations. Higher order wave evolution and wave-wave interaction are implicitly accounted for as well as trough to crest asymmetries, wave skewness, and even wave breaking, provided the discretization is sufficient (Oberhagemann et al., 2012). However, initial and boundary conditions impose a wave regime according to eq. (9). Long simulation times and large fluid domains may be required to yield a fully developed wave process, including all nonlinearities. More advanced boundary conditions may help. The simplest model that describes the weakly nonlinear evolution of a narrow band,

unidirectional wave system in deep water is the Nonlinear Schrödinger (NLS) equation:   A 1  A  k0  2 A 2  i   k0 A A  0   x c t   2 t 2 g 0  

(10)

Here, A(x,t) describes the complex envelope of the waves and is related to surface elevation ζ(x,t):

 ( x, t )  A( x, t ) cos(k0 x   0t ) ,

(11)

Tab. 1: Main particulars of subject ships

Length overall Length bet. per. Molded breadth Draft Block coeff. Displacement

Containership 1

Containership 2

349.00 m 333.44 m 42.80 m 13.10 m 0.62 125604 t

336.60 m 321.00 m 48.40 m 15.00 m 0.62 143742 t

where k0 is the wave number corresponding to the dominant wave and ω0 = ω(k0) the corresponding angular frequency. Equation (10) describes dynamics of waves in a quasi-linear regime properly, and on average it can decently reproduce statistical properties of surface elevation and wave height. The nonlinear evolution of waves is expected to become important for increasing sea state steepness and narrowing bandwidth, i.e. increasing γ. Steep sea states are most contributing to extreme ship responses, and relatively high values of γ may be representative for these conditions due to limited fetch and duration. The nonlinear evolution of waves and the formation of rogue wave groups take space to develop. A rough estimate is a length of the order of 30 characteristic wave lengths (Onorato et al. 2006). Instead of using large solution domains in RANS simulations, pre-simulations with a fast NLS solver is an attractive alternative. Fig. 1: Model testing of containerships 1 and 2 MODEL TESTS AND VALIDATION Model tests of two large modern containerships were carried out in the basin of Canal de Experencias Hidrodinámicas Del Pardo in Madrid (CEHIPAR) and the Korean Research Institute for Ships and Offshore (KRISO) to validate the developed numerical methods. Tab. 2 lists principal particulars of these two ships. To measure sectional loads, the containership 1 and 2 models consisted of six segments (Fig. 1). Both containership models were equipped with a backbone that reflected the basic vibration modes and natural frequencies of the full scale ship. Model tests comprised runs in regular waves and irregular long-crested waves. Prerequisite linear seakeeping computations determined relevant and ship specific wave and sea state parameters. In addition to irregular waves obtained from random realizations of sea states, dedicated deterministic sequences were investigated that aimed at producing ship responses of given magnitudes.

Fluid dynamic computations covered a large number of simulations for various wave conditions, for which a large number of numerical grids were generated. Common to all grids was the refinement region in front of the ship near the free surface to avoid loss of wave energy due to insufficient grid resolution. The vertical extension of this refinement box depended on the significant wave height and mean wave period. A symmetry boundary condition was applied at the vertical center plane for head wave conditions. At the pressure outlet of the fluid domain, cells were stretched, and a numerical beach consisting of sources terms was placed to dampen the waves. The meshes consisted of approximately 500,000-800,000 control volumes. Figure 2 illustrates the mesh for the containership 1, including typical domain extensions as multiples of ship length. Grid studies were undertaken to estimate the discretization errors. The time step size per wave period was adjusted to abide with appropriate Courant numbers.

Fig. 2: Computational field domain boundaries Structural Finite Element models generated for the two ships used shell and beam elements. Two different structural representations were investigated based on the Timoshenko-Beam approach and the three-dimensional FE model approach, respectively. TIMOSHENKO BEAM MODEL Eigenvalue analysis of the Timoshenko beam models yielded the natural frequencies of vertical bending vibration and associated mode shapes. Figure 3 exemplarily shows modes and associated dry frequencies of the containerships. The flexural properties of beam elements complied with the experimental models. Frequencies, here given in Hz, refer to full-scale values. Element attributes (profile, material and mass properties, etc.) were calibrated to match the natural frequencies of the tested models in the dry condition. Free vibration decay simulations were performed with the FE solver coupled to the two fluid dynamic codes COMET and the newly developed coupled free-surface solver from the CFD-toolbox OpenFOAM. Vibrations, initially excited due to imbalance of mass and floatation, decayed freely so the dominating natural vibration frequency for the wetted hull surface and the related damping ratio could be determined, see Fig. 4. The two-node vibration mode was dominant, and higher frequencies observed at the beginning of the simulation decayed rapidly. Both coupled fluid solvers obtained basically the same vibration frequency and same damping ratio. Tab. 3 lists the frequencies obtained for the two-node vertical bending mode. Damping ratios were determined for the Timoshenko beam approach only.

Fig. 3: Mode shapes and associated natural frequencies resulting from eigenvalue analysis of containerships 1 (top) and 2 (bottom) Tab. 2: Two-node vibration frequencies [s−1] and damping ratios for the containership 1 (4 top rows) and for the containership 2 (2 bottom rows) Model

Dry ship

Wet ship

damping ratio

Beam (COMET) Beam (OpenFOAM) 3D-FE model Experiments Beam (COMET) Experiment

0.673 0.672 0.682 0.673 0.603 -

0.520 0.519 0.477 0.514 0.43 0.43

0.0095 0.0098 − 0.016 0.020 0.019

Natural frequencies of the ships afloat were significantly lower than dry natural frequencies due to effects of hydrodynamic added mass. While the ratio of wet to dry natural frequency of containership 2 is typical in that it suggests the added mass to be approximately the same as the ship’s mass, results for containership 1 suggest a significantly smaller effect of added mass, both in experiments and numerical results. Damping, consisting of hydrodynamic damping, was assessed in grid studies of free vibration decay simulations (el Moctar et al., 2011). Here, structural damping was set to zero.

Fig. 4: Time series of normalized vertical vibration acceleration from free vibration decay simulations of containership 1 at Fn=0.07

Fig. 6: Entire FE model of the containership, including acoustic elements located inside a hemispherical fluid domain

Fig. 7: Containership 1, mode shape for two-node vertical bending of the wet hull

Fig. 5: Time series of normalized vertical bending moment from free vibration decay simulations of containership 2 at Fn=0.0 Although initially somewhat higher acceleration oscillations resulted (for the containership 1) using coupled simulations with OpenFOAM, see Fig. 8, results obtained with both fluid solvers generally compared favorably. THREE-DIMENSIONAL FE MODEL A 3D FE-model was generated for the containership 1 and was used to assess the ability of the applied numerical methods to reliably predict global and local loads. In contrast to the Timoshenko beam model, the 3D-FE model allows to also account for torsional deformations. The FE model of the containership was arranged in compliance with the physical model, including an aluminum backbone. Moreover, the numerical model consists of connecting links and mass elements. Apart from mass elements, shell and beam elements were weightless. Stiff link elements transmitted axial loads from the backbone to the hull, and link pairs transferred moments.

To ensure a realistically low shear center, the backbone was positioned at the lowest position. Geometric restrictions required the forepart of the backbone to be sloped. The ends of the backbone were clamped to the hull. Figure 6 presents the entire FE model, including acoustic elements located inside a hemispherical fluid domain of diameter equal to 1.3 times the overall length of the containership. Comparatively high stiffness shell plating idealized the hull. To allow for global deformations, five gaps were provided to separate the shell plating from the hull. The backbone thickness was kept constant. To minimize the number of elements and nodes, element sizes increased with distance from amidships. Modal analyses were performed to facilitate the arrangement of rigid links. Natural frequencies and the mode shapes of the basic hull girder vibrations were analyzed. As a sample result, Fig. 7 shows the mode shape for the two-node vertical bending of the wet hull. SHIP RESPONSE IN SEVERE SEAS Adequate RAOs are a premise for reliable predictions of ship responses. Comparisons of RAOs between numerical and model test results revealed satisfying agreements. Indeed, linear responses shall not be

focused here. However, before analyzing the feasibility of numerical methods to predict ship responses in extreme natural seaways and wave trains, nonlinear regular waves with significant wave heights and steepness were investigated using the different coupled solvers. Here, we present exemplary results for containership 1 in two regular waves with wave height H=8.0 m and wave periods T=11.65 s and T=9.65 s (Ley and el Moctar, 2014), respectively. The ratios between the wet eigenfrequency wwet of the dominant two node vertical bending moment mode and the angular encounter frequency we are 3 and 4, respectively. Dynamic effects in vertical bending moment were pronounced, see Fig. 8. The green curve represents the results for the three-dimensional FEmodel based on the 1-way coupling method, enhanced by acoustic elements, to account for added mass effects. The time axis is scaled by the encounter period te of the wave. Phasing of the vertical bending moment based on the 3D-FE model agrees well with experiments and the beam model, see Fig. 8.

Fig. 8: Containership 1 sailing with vs=15 kts in regular waves with H=8.0 m and T=11.65 s (top) and H=8.0 m and T=9.65 s (bottom), scaled vertical bending moment My, comparison of threedimensional FE-model and beam model Although the fundamental wet natural frequency of the 3D FE model is smaller compared to the other models, the time series of vibrations are fairly captured. Corresponding structural responses obtained by neglecting inertia effects (quasi-static)

are included as well, demonstrating the importance of dynamic effects for this wave condition. The beam model underestimated the vibration amplitudes in this second case, while the first case shows a good agreement. Simulations in irregular sea states provide statistical information about ship responses. For linear wave and ship response processes, all statistical information about the ship response to irregular waves is readily available from spectral moments. Rice’s formula and assuming narrow-banded response spectra allow efficient calculation of response probability distributions in closed form expressions. The situation is more complex for nonlinear responses, especially when vibratory modulations of the response are present. Cost intensive simulations in the time domain using random wave components may then become the only choice to obtain reliable information about response probability distributions. Numerical results presented here are based on the Timoshenko-Beam approach. They not only validate the numerical methods, they also help to assess the feasibility of using transient RANS methods to obtain short-term statistical characteristics of nonlinear ship response. Investigations focus on the vertical hull girder bending moment amidships, My, which is a key parameter in ship design.

Fig. 9: Sample time histories of normalized midship vertical bending moment for the containership in seaway with Hs = 12.5 m and Tp = 11.8 s Fig. 9 presents sample time histories of numerical and experimental results of the midship vertical bending moment My, normalized with ρgBLpp², for the containership 1 in irregular longcrested head waves with Hs = 12.5 m and Tp = 11.8 s. The time records revealed significant effects of vibration caused by bow flare slamming events. E.g., the black dot marks a time instant of slamming impact. In the computations, surge motions were

imposed on the ship according to the measured surge motions, and wave elevations were reproduced from wave probe measurements ahead of the ship. Overall agreement is fine. Vibratory amplitudes after the impact fairly match, and the vibration frequency is well reproduced numerically. Time records of 1000 to 1400s duration of the flexible containership 1 model were available for comparison, from numerical computations as well as model tests. Range-pair counting yielded exceedance rates of ship response cycles. Figure 10 (top) shows the exceedance rates of vertical bending moments, evaluated from time series samples of 1400 s length. Low-pass filtering with a cut-off frequency of 0.25 Hz eliminated the high-frequent vibratory part from time histories to highlight the contribution of vibration. The remainder was associated with rigid body vertical bending moment. The slopes of exceedance rates differ significantly for the filtered signal because, first, hull girder vibrations substantially increased the number of response cycles. This was favorably replicated in numerical simulations.

at 10 knots in a long-crested sea state with Tz = 11.5 s, Hs = 14.5 m, γ = 1.0, Fig. 11 shows the resulting amplitudes of normalized vertical bending moment, plotted against the number of exceedances. Rainflow counting evaluated time series of 5000 s. Additional time series of 5000 s were produced, each of which accounted for one of the following: reducing the ship speed to vS = 5 kts; using a rigid hull girder; using cosine square wave spectral spreading instead of long-crested waves. The rigid hull girder simplification strongly reduced not only the overall number of encountered load cycles, but also the number of cycles exceeding a given load level.

Fig. 11: Comparative normalized amplitude distributions of midship vertical bending moment for containership 2 in sea state with Tz = 11.5s, Hs = 14.5m, γ = 1.0, v=10kts

Fig. 10: Exceedance rates of range-pair counted normalized midship vertical bending moment cycles for the containership 1 Second, the maximum response cycle obtained from the unfiltered signal was about 60 percent larger, compared to the filtered signal. Numerically and experimentally determined exceedance rates basically show the same vibratory amplification, but numerical results tend to under-predict the responses at low exceedance rates where statistical uncertainty is largest. In general, numerical results agreed fairly well with experimental measurements. For containership 2, we exemplified the impact of ship speed, wave directional spreading, and hull girder flexibility on the probability distributions of vertical bending moment. For the ship advancing

This finding emphasized the importance of hull girder vibration on vertical bending moment. The speed reduction from 10 kts to 5 kts apparently had only a small influence on loads, although bow flare slamming was less pronounced at the lower speed. However, stern slamming becomes an issue. The directional spreading of waves caused a comparable decrease of the loads over the greater part of the cumulative distribution. Next, we performed RANS simulations to investigate the containership’s midship vertical bending moment, My, in the two different sea state conditions A and B with parameters listed in Tab. 3. The ship had zero speed and the waves consisted of long-crested head waves. Significant wave height, Hs, was the same for both sea states; however, peak period, TP, zero up-crossing period, Tz, and steepness, γ, differed. For both conditions, we obtained the wave sequence phasing randomly as well as from the NLS method. The evaluated time duration of our RANS

simulations comprised only of 2500 s. Example time series of midship vertical bending moments are shown in Fig. 12. Figure 13 presents cumulative exceedance distributions found from rainflow counting of the My time series. Especially for larger values of My and the steeper sea state condition B with γ = 6.0, differences are remarkable. Tab. 3: Parameters of seaways A and B Seaway

Tp

Tz

Hs

γ

A B

16.2 s 15.0 s

11.5 s 12.4 s

14.5 m 14.5 m

1.0 6.0

This indicated a significant influence of wave process evolution on hull girder loads in these severe seaways, particularly for a narrow-banded spectrum with γ = 6.0. CONDITIONED WAVE SEQUENCES Determining lifetime maximum values of a ship response through direct long-term simulations is impossible with complex numerical methods because of typical life cycles of 20 to 25 years. Instead, short sequences are sought that represent the expected lifecycle maximum response, i.e. wave events tailored to produce this response.

+

Fig. 14 Example MLRW wave elevation as a function of time and distance from target location Fig. 12: Comparative surface elevation (top) and midship vertical bending moment (bottom) for the containership1 in seaway B with random phasing (dashed curve) and with phasing from the NLS method (solid lines)

Fig. 13: Comparative rainflow counted exceedances of midship vertical bending moment amplitudes for the containership 2 in seaways A and B

Here, we used response-conditioned wave sequences according to the concept of the Most Likely Response Wave, (Dietz, 2004). Based on earlier developments to construct wave sequences of a prescribed maximum wave elevation, Dietz proposed a method to condition a wave sequence to cause a given response based on the linear transfer functions of that response and a discretised wave energy spectrum. The underlying idea is a spectral moment transformation of the discretized linear response spectrum, resulting in a wave sequence that causes a given magnitude of the response at a given time instant. The associated linear ship response is proportional to the autocovariance function of its spectrum, and corresponds to the expected (i.e. statistical mean) shape of a response of the target amplitude from random simulations. A nonlinear simulation in the wave sequence then provides a nonlinear correction of the linear response. MLRW wave sequences are transient wave packets of short duration and limited spatial extent, see Fig. 14 for an example. This makes them attractive for RANS CFD simulations since required computing times are very short, e.g. Seng (2012). Moreover, the wave elevation far off the target location is well captured with linear theory, so the

boundary conditions may use linear wave descriptions regardless of the wave steepness at the target location. Fig. 15 shows, for the containership 2, time series of simulations in a MLRW sequence. The response, here the ship’s midship vertical bending moment in sagging, is normalized against the target value. The dotted line corresponds to the linear response which was used for constructing the wave sequence, while the other time series result from a CFD simulation using a Timoshenko beam model of the ship hull girder, and an additional comparative computation for the rigid hull. The target time was set to t0 = 60 s to avoid biases caused by initial conditions. Per definition, the corrected response should occur at the target time. However, nonlinearities presumably affected not only response amplitudes, but also response times. Therefore, the peak of nonlinear responses close to target time was evaluated instead. Close to target time, RANS simulations deviate most from linear predictions, while they resemble linear predictions far away from target time. Observed differences are associated with the nonlinearities of the response and the wave process itself. Significant nonlinearities of the response are hogging / sagging asymmetry and vibration initiated from slamming impacts.

reported increasing discrepancies for small exceedance rates in comparisons with Monte Carlo simulation (MCS) in random irregular waves. A reason for the discrepancy is that, although MLRW wave sequences are the statistical mean of a wave sequence causing a linear response of given magnitude, this is not necessarily the case for the nonlinear response. An alternative can be conditioned wave sequences, embedded in random background waves, so-called Conditioned Random Response Waves (CRRW). Instead of a single simulation for each response magnitude, a representative set of CRRW sequences is then required to establish probability distributions of the nonlinear response, conditional on the linear response. Unconditioning with respect to the (known) linear probability distribution may be used in a next step to obtain the nonlinear distribution function (Dietz, 2004). This approach, however, increases the number of required simulations by one to two orders of magnitude. Drummen et al. (2012) demonstrated fine agreement with exceedance probability functions from random irregular wave realizations. Fair agreement was achieved a simplified approach that relied on averaging over a smaller set of CRRW simulations, see also Oberhagemann et al. (2012) for a similar study with the present CFD method, also for containership 2. Both MLRW and simplified CRRW approach underpredicted exceedance rates. Results of a more accurate evaluation of CRRW simulations are presented in Fig. 16. Nonlinear exceedance rate distributions were obtained from unconditioning based on the original proposal of Dietz (2004).

Fig. 15 Time series of midship vertical bending moment for the containership in an MLRW sequence The nonlinear response peak at target time is interpreted as the corrected linear response, and it is assumed to be associated with the same probability of exceedance. Nonlinear simulations for a set of different response magnitudes should then allow efficient estimates of nonlinear short-term probability functions. Drummen et al. (2012) tested the feasibility of establishing short-term probability functions of vertical bending moments from MLRW sequences numerically and experimentally. They

Fig. 16 Short-term exceedance rates of vertical bending moment from linear seakeeping, random waves, and conditioned wave sequences

Still, differences are observed between distributions from random irregular wave simulations and the CRRW approach, but smaller than for MLRW simulations. These fail to predict the response magnitude already for large exceedance probabilities, and rather resemble the distribution function for the rigid ship found from MCS. Wave sequences were generated without accounting for wave-induced vibration. Vibration observed during coupled nonlinear simulations is a random effect of nonlinearity, and its magnitude changes with the encounter frequency of the incident wave sequence. Simulations in wave sequences representing shorter peak periods of the underlying wave spectrum resulted in significantly increased vibration amplifications. SUMMARY AND CONCLUSIONS Numerical nonlinear time domain simulation methods assessed wave-induced structural loads of two containerships in head seas. Two different approaches of modeling extreme irregular seaways were introduced, namely, the traditional procedure based on the superposition of random harmonic wave components and on a procedure that requires a presimulation of the wave field by solving the Nonlinear Schrödinger equations. Numerical methods were applied that account for hull girder flexibility. The first approach iteratively two-way coupled a Timoshenko beam model based on the finite element method to the flow solver in order to replicate the global stiffness of the ship structure. The second approach considered a full three-dimensional finite element model. The two-way coupling approach relied on in-house libraries of mapping routines and structure solvers which were linked to the RANS solvers COMET and interDyMFoam, while MpCCI was used to couple the three-dimensional FE solver Ansys to the RANS solver StarCCM+, i.e. map the loads from the CFD mesh onto the FE mesh and return structural displacements to the CFD solver. Additional routines were implemented to balance residual forces resulting from the load transfer. This was necessary to suppress rigid body motions in the transient structural analysis. Results were presented for an enhanced one-way coupling method to transfer wave loads onto the FE model and for a weak two-way coupling technique. For the oneway coupling method, the structural FE model was supplemented by fluid elements (acoustic elements) attached to the submerged hull shell. These elements accounted for important transient effects of

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